These notes characterize maxima and minima in terms of first derivatives. I focus primarily on maximization. The problem of minimizing a function f has the same solution (or solutions) as the problem of maximizing −f , so all of the results for maximization have easy corollaries for minimization. The main result of these notes is the Karush-Kuhn-Tucker (KKT) Theorem, recorded as Theorem 3 in Se...

In this note we give an elementary proof of the Fritz-John and Karush–Kuhn–Tucker conditions for nonlinear finite dimensional programming problems with equality and/or inequality constraints. The proof avoids the implicit function theorem usually applied when dealing with equality constraints and uses a generalization of Farkas lemma and the Bolzano-Weierstrass property for compact sets. 2006 P...

In this paper we characterize the local maxima of a continuous global optimization formulation for finding the independence number of a graph. Classical Karush-Kuhn-Tucker conditions and simple combinatorial arguments are found sufficient to deduce several interesting properties of the local and global maxima. These properties can be utilized in developing new approaches to the maximum independ...

The author regrets that some terms are missing or misspecified in the Karush–Kuhn–Tucker conditions in Sections 3.7–3.10 on page 346. Corrections are given below. add multiplier p m c m (discount rate in tree node m multiplied by the probability of its occurrence) to d d (number of days in demand season d).

We study nonsmooth multiobjective programming problems involving locally Lipschitz functions and support functions. Two types of Karush-Kuhn-Tucker optimality conditions with support functions are introduced. Sufficient optimality conditions are presented by using generalized convexity and certain regularity conditions. We formulate Wolfe-type dual and Mond-Weirtype dual problems for our nonsmo...

In this paper, the common centralized DEA models are extended to the bi-level centralized resource allocation (CRA) models based on revenue efficiency. Based on the Karush–Kuhn–Tucker (KKT) conditions, the bi-level CRA model is reduced to a one-level mathematical program subject to complementarity constraints (MPCC). A recurrent neural network is developed for solving this one-level mathematica...